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Statistical properties of spectral fluctuations for a

quantum system with infinitely many components

H. Makino1, N. Minami2, and S. Tasaki3

1Department of Human and Information Science, Tokai University, 1117

Kitakaname, Hiratsuka-shi, Kanagawa 259-1292, Japan2School of Medicine, Keio University, Hiyoshi 4-1-1, Kohoku-ku,

Yokohama, Kanagawa 223-8521, Japan3Department of Applied Physics, Waseda University, 3-4-1 Okubo,

Shinjuku-ku, Tokyo 169-8555, Japan

Abstract

Extending the idea formulated in Makino et al[Phys.Rev.E 67,066205], that is

based on the Berry–Robnik approach [M.V. Berry and M. Robnik, J. Phys. A 17,

2413], we investigate the statistical properties of a two-point spectral correlation for

a classically integrable quantum system. The eigenenergy sequence of this system is

regarded as a superposition of infinitely many independent components in the semi-

classical limit. We derive the level number variance (LNV) in the limit of infinitely

many components and discuss its deviations from Poisson statistics. The slope of the

limiting LNV is found to be larger than that of Poisson statistics when the individual

components have a certain accumulation. This property agrees with the result from the

semiclassical periodic-orbit theory that is applied to a system with degenerate torus

actions[D. Biswas, M.Azam,and S.V.Lawande, Phys. Rev. A 43, 5694].

1 typeset using PTPTEX.cls 〈Ver.0.9〉

http://arxiv.org/abs/0902.1282v2

§1. Introduction

Study of energy level statistics has played an important role in elucidating the universal

properties of quantum systems, which in the semiclassical limit, reflect regular and chaotic

features of classical dynamics. Berry and Tabor conjectured that the eigenenergy levels of a

quantum system, whose classical dynamical system is integrable, have the same fluctuation

property as uncorrelated random numbers from a Poisson process, and thus are character-

ized by Poisson statistics.1) This conjecture is in contrast with the conjecture of Bohigas,

Giannoni, and Schmit, which assert that GOE or GUE statistics based on random matrix

theory (RMT) are applicable to the fluctuation property of energy levels of a quantum sys-

tem whose classical dynamical system is fully chaotic.2) These contrasting conjectures have

been examined using various statistical observables, e.g., the nearest-neighbor level-spacing

distribution (NNLSD) and the level number variance (LNV).3)

The NNLSD P (S) is the observable most commonly used to study short-range fluctua-

tions in a spectrum. For eigenenergy levels on the unfolded scale,4) this observable is defined

as the probability density of finding a distance S between adjacent levels. In Poisson statis-

tics, it is characterized by the exponential distribution P (S) = e−S; on the other hand, in

GOE/GUE statistics, it approximates the Wigner distribution very well.3)

The LNV Σ2(L) is the observable commonly used to study correlations between pairs of

levels, which characterizes long-range fluctuations in a spectrum. It is defined as the average

variance of the number of levels in an energy interval containing an average of L levels. On

the unfolded scale, this interval is equivalent to an interval of length L, and the LNV is thus

defined as

Σ2(L) = 〈(N (ǫ+ L)−N (ǫ)− L)2〉, (1.1)

where N (ǫ) is the number of eigenenergy levels below ǫ, and the brackets 〈· · · 〉 represent

the average over the value of ǫ. In Poisson statistics, the LNV is equal to the number it-

self [i.e., Σ2Poisson

(L) = L]; on the other hand, in RMT, in which β = 1 and 2 correspond to

the GOE and GUE level statistics, respectively, the LNV increases logarithmically with L[

i.e., Σ2β(L) ∼ (2/βπ2) ln(2πL)

]

.3) According to Berry’s semiclassical theory,5) the LNV of

a quantum system with f degrees-of-freedom should display these universal statistics in the

region L

to be characterized by Poisson statistics is now widely accepted as a universal property of

generic integrable quantum systems in the semiclassical limit.

One possible mechanism producing Poisson statistics has been proposed by Makino et

al ,19) based on the Berry-Robnik approach.20) We briefly review the outline below: For an in-

tegrable system, individual orbits are confined in each inherent torus whose surface is defined

by holding its action variable constant, and the whole region of the phase space is densely

covered with invariant tori. In other words, the phase space consists of infinitely many re-

gions, which have infinitesimal volumes in Liouville measure. Because of the suppression of

quantum tunneling in the semiclassical limit, the Wigner functions of quantal eigenstates are

expected to be localized in the phase space region explored by a typical trajectory,21), 22) and

to form independent components. For a classically integrable quantum system, the Wigner

function localizes on the infinitesimal region in ~ → 0 and tends to a δ function on a torus

that is designated by quantum numbers.23) Then, the eigenenergy levels can be represented

as a statistically independent superposition of infinitely many components, each of which

gives an infinitesimal contribution to the level statistics. Therefore, if the individual spectral

components are sparse enough, one would expect Poisson statistics to apply here, as a result

of the law of small numbers.24)

The statistical independence of spectral components is assumed to be justified by the

principle of uniform semiclassical condensation of eigenstates in the phase space and by the

lack of their mutual overlap, and thus, can only be expected in the semiclassical limit.22), 25)

This mechanism was introduced as a basis for the Berry-Robnik approach to investigate

the relation between the phase space geometry and the formation of energy level sequence

in a generic mixed quantum system, whose classical dynamics is partly regular and partly

chaotic.20) The validity of the Berry-Robnik approach is confirmed by numerical computa-

tions for the mixed quantum systems in the extremely deep semiclassical region,26), 27) which

is called the Berry-Robnik regime.28)

Based on this view, Makino and Tasaki investigated the short-range spectral statistics

of classically integrable systems.19) They derived the cumulative function of NNLSD, i.e.,

M(S) =∫ S

0P (x)dx, is derived in the limit of infinitely many components, which is charac-

terized by a single monotonically increasing function µ̄(0;S) of the nearest level spacing S

as

Mµ̄(S) = 1− [1− µ̄(0;S)] exp

(

−

∫ S

0

[1− µ̄(0; x)] dx

)

. (1.2)

The function µ̄(0;S) classifies Mµ̄(S) into three cases: Case 1, Poisson distribution Mµ̄(S) =

1 − e−S for ∀S ≥ 0 if µ̄(0; +∞) = 0; Case 2, asymptotic Poisson distribution, which con-

verges to the Poisson distribution for S → +∞, but possibly not for small spacings S if

3

0 < µ̄(0; +∞) < 1; and Case 3, sub-Poisson distribution, which converges to 1 for S → +∞

more slowly than does the Poisson distribution, if µ̄(0; +∞) = 1. Therefore, the Berry-

Robnik approach when applied to classically integrable quantum systems, admits deviations

from Poisson statistics.

Cases 2 and 3 are possible when the eigenenergy levels of individual components show

a singular NNLSD result from strong accumulation19) (see also Section 3 of the present

paper), which is expected when the classical dynamical systems have a spatial symme-

try1), 11), 14), 17), 18), 29), 30), 31), 32) or a time-reversal symmetry.33), 34), 35) One possible example

is a rectangular billiard with a rational ratio of squared sides.1), 11), 18), 31), 32) These results

suggest the existence of a new statistical law different from Poisson statistics in the strongly-

degenerate quantum systems, and raises the question as to whether similar behaviors appear

in the two-point spectral correlation.

Deviation from Poisson statistics, resulting from the symmetry, is also exhibited within

the framework of the periodic-orbit theory.36) Based on the semiclassical theory of Berry

and Tabor,7) Biswas, Azam, and Lawande investigated the LNV for classically integrable

systems with degeneracy in orbit actions.32) They showed that the slope gav(L) of the LNV

Σ2(L) = gav(L)L is described by the average degeneracy of actions of the periodic orbits,

which for the degenerated quantum systems, is greater than 1, and it is the slope of the

variance of Poisson statistics. This result leads to the possibility that the degeneracy of ac-

tions, induced by the symmetry, could be an essential factor for the singularity of individual

spectral components that yields Cases 2 and 3. This possibility is confirmed by examining

the properties of LNV for Cases 2 and 3.

In this study, extending the theory of Makino and Tasaki,19) we investigate the LNV

Σ2(L) of quantum systems whose energy levels consist of infinitely many independent com-

ponents, and show that the non-Poisson limits (Cases 2 and 3) are possibly observed also in

the long-range spectral fluctuations.

Based on the Berry-Robnik approach, the overall LNV is derived as follows: We con-

sider a system whose classical phase space is decomposed into N disjoint regions that give

the distinct spectral components. The Liouville measures of these regions are denoted by

ρn(n = 1, 2, 3, · · · , N), which satisfy∑N

n=1 ρn = 1. Let E(k;L), k = 0, 1, 2, · · · be the distri-

bution function, which denotes the probability to find k levels in an interval (0, L).3), 37), 38)

The LNV Σ2(L) is expressed by E(k;L) as

Σ2(L) =+∞∑

k=0

(k − L)2E(k;L). (1.3)

4

Let P (k;S), k = 0, 1, 2, · · · , be the level-spacing distribution, which denotes the probability

density to find k levels in an interval of length S beginning at an arbitrary level ǫi and ending

at the level ǫi+k+1. P (k;S) is related to E(k;L) as

P (k;L) =∂2

∂L2

k∑

j=0

(k − j + 1)E(j;L), (1.4)

and

E(k;L) =

∫ +∞

L

dx

∫ +∞

x

[P (k;S)− 2P (k − 1;S) + P (k − 2;S)] dS, (1.5)

where P (j < 0;S) = 0.3), 37), 38) Eqs.(1.4) and (1.5) are known as the formulae in the the-

ory of point process, and are derived as corollaries of the Palm-Khinchin formula.39) The

NNLSD, which was expressed by P (S) at the beginning of this section, is the special case

with k = 0.

When the entire sequence of energy levels is a product of statistically independent su-

perposition of N subsequences, E(k;L) is decomposed into those of subsequences, en(k;L)

as3), 40)

EN(k;L) =∑

PNn=1 kn=k

N∏

n=1

en(kn; ρnL), (1.6)

where en satisfies the normalization conditions

+∞∑

k=0

en(k; ρnL) = 1 (1.7)

and+∞∑

k=0

ken(k; ρnL) = ρnL. (1.8)

In terms of normalized level-spacing distribution pn(k;S) of the subsequence, en(k;L) is

specified as

en(k;L) = ρn

∫ +∞

L

dx

∫ +∞

x

[pn(k;S)− 2pn(k − 1;S) + pn(k − 2;S)] dS, (1.9)

where pn(j < 0;S) = 0, and pn(k;S) satisfies the normalizations

∫ +∞

0

pn(k;S)dS = 1 (1.10)

and∫ +∞

0

Spn(k;S)dS =k + 1

ρn. (1.11)

5

Note that in general, the individual components are not always unfolded automatically even

when the overall spectrum is unfolded. However, in a sufficiently small interval [ǫ, ǫ +∆ǫ],

each spectral component obeys the same scaling law (see Appendix of Makino et al19)) and

thus is unfolded automatically by an overall unfolding procedure. In the Berry-Robnik ap-

proach, Eq.(1.6) relates the level statistics in the semiclassical limit with the phase space

geometry.

In most general cases, the individual components might have degeneracy of levels that

lead to singular level spacing distributions. In such a case, it is convenient to use its cumu-

lative distribution function µn(k;S),

µn(k;S) =

∫ S

0

pn(k; x)dx. (1.12)

This function satisfies

µn(k − 1;S) ≥ µn(k;S) for all k ≥ 1 and for all S ≥ 0, (1.13)

and the normalization condition

+∞∑

k=0

[µn(k − 1;S)− µn(k;S)] = 0, (1.14)

with µn(j = −1;S) = 0, since fn(k;S),

fn(k;S) =

{

µn(k − 1;S)− µn(k;S), k ≥ 1

1− µn(0;S), k = 0

denotes the probability to find k levels in an interval of length S beginning at an arbitrary

level, and obviously satisfies, from the definition of this function, the conditions fn(k;S) ≥ 0

and∑+∞

k=0 fn(k;S) = 1.

In addition to Eq.(1.6), we introduce the following two assumptions that were introduced

in Ref.:19)

Assumption (i). The statistical weights of independent regions vanish uniformly in the

limit of infinitely many regions:

maxn

ρn → 0 as N → +∞. (1.15)

Assumption (ii). For k = 0, 1, 2, · · · , the weighted mean of the cumulative distribution of

energy spacing, namely,

µ(k; x) =N∑

n=1

ρnµn(k; x) (1.16)

6

converges as N → +∞ to µ̄(k; x),

limN→+∞

µ(k; x) = µ̄(k; x), (1.17)

where the convergence is uniform on each closed interval: 0 ≤ x ≤ S. In the Berry-Robnik

approach, the statistical weights of individual components are the phase volumes (Liouville

measures) of the corresponding invariant regions.

Under the Assumptions (i) and (ii), Eqs.(1.3) and (1.6), we obtain the overall LNV in

the limit N → +∞:

Σ2µ̄(L) = L+ 2

∫ L

0

+∞∑

k=0

µ̄(k;S)dS. (1.18)

When µ̄(k;S) = 0 for all k, the LNV of the whole energy sequence reduces to Σ2Poisson

(L) =

L. This condition is expected when the individual components are sufficiently sparse. In

general, we expect µ̄(k;S) > 0, which corresponds to a certain accumulation of the levels of

individual components. In this case, the LNV of the whole energy sequence deviates from

Σ2Poisson

(L) in such a way that the slope of the variance is greater than 1.

The present paper is organized as follows. The limiting LNV (1.18) is derived from

Eqs.(1.3) and (1.6), and Assumptions (i) and (ii) in Section 2. In Section 3, the property of

the limiting LNV is analyzed for the Cases 1–3, wherein the deviations from Poisson statistics

are clearly observed in Cases 2 and 3. We present a numerical analysis for the rectangular

billiard that shows deviations from Poisson statistics in Section 4. In the concluding section,

we discuss some relationships between our results and other related works.

§2. limiting level number variance

In this section, by using Eqs.(1.6) and (1.3), and Assumptions (i) and (ii) introduced in

Section1, we derive the limiting LNV,

Σ2µ̄(L) = L+ 2

∫ L

0

+∞∑

k=0

µ̄(k;S)dS, (2.1)

in the limit of infinitely many components N → +∞.

First, we rewrite Eq.(1.3) in terms of the function en, and decompose it into the LNV σ2n

of individual components:

Σ2N(L) =

+∞∑

k=0

(k − L)2∑

PNn=1 kn=k

N∏

n=1

en(kn; ρnL) (2.2)

=N∑

n=1

+∞∑

k=0

(k − ρnL)2en(k; ρnL) ≡

N∑

n=1

σ2n(ρnL), (2.3)

7

where we have used the properties∑+∞

k=0EN (k;L) = 1,∑+∞

k=0 kEN(k;L) = L, and

+∞∑

k=0

k2EN (k;L) =N∑

n=1

+∞∑

kn=0

k2nen(kn; ρnL) + L2 −

N∑

n=1

(ρnL)2, (2.4)

which follow from Eqs.(1.7) and (1.8), and the relationships k =∑N

n=1 kn and∑N

n=1 ρn = 1

(see also Appendix A). The functions en(k; ρnL) in the above equation are rewritten in terms

of the cumulative level-spacing distribution functions µn(k;S) of individual components.

en(k; ρnL) =

−ρn∫ +∞

LdS [µn(k;S)− 2µn(k − 1;S) + µn(k − 2;S)] , k ≥ 2

−ρn∫ +∞

LdS [µn(1;S)− 2µn(0;S) + 1] , k = 1

ρn∫ +∞

LdS [1− µn(0;S)] , k = 0

(2.5)

=

ρn∫ L

0dS [µn(k;S)− 2µn(k − 1;S) + µn(k − 2;S)] , k ≥ 2

ρn∫ L

0dS [µn(1;S)− 2µn(0;S) + 1] , k = 1

1− ρn∫ L

0dS [1− µn(1;S)] , k = 0

(2.6)

where Eq.(2.6) follows from Eq.(1.11), integration by parts and the following limits which

result from the existence of the average:

limS→+∞

S [µn(k;S)− 2µ(k − 1;S) + µn(k − 2;S)] = 0, k ≥ 2

limS→+∞

S [µn(1;S)− 2µ(0;S) + 1] = 0, k = 1 (2.7)

limS→+∞

S [1− µn(0;S)] = 0, k = 0

Then, σ2n(ρnL) is described in terms of µn(k;S) as

σ2n(ρnL) =+∞∑

k=0

k2en(k; ρnL)− ρ2nL

2

= ρn

∫ L

0

dS

+∞∑

k=1

(2k + 1) [µn(k − 1;S)− µn(k;S)] + ρn

∫ L

0

[1− µn(0;S)] dS − ρ2nL

2

= ρnL+ 2

∫ L

0

dS

+∞∑

k=0

ρnµn(k;S)− ρ2nL

2, (2.8)

where we have used Eq.(1.14) and the relation

+∞∑

k=0

[(k − 1)µn(k − 1;S)− kµn(k;S)] = 0 (2.9)

with µn(−1;S) = 0. Therefore, in the limit N → +∞, we have the convergence

Σ2µ̄(L) = limN→+∞

N∑

n=1

σ2n(ρnL) = L+ 2

∫ L

0

dS+∞∑

k=0

µ̄(k;S), (2.10)

8

where the limit (2.10) follows from Assumption (ii) and the following property results from

Assumption (i):N∑

n=1

ρ2n ≤ maxn

ρn

N∑

n=1

ρn = maxn

ρn → 0. (2.11)

We note the spectral rigidity ∆3(L), which was introduced by Dyson and Mehta.41)

In combination with the LNV, this quantity has played a major role in the study of the

long-range spectral statistics. According to Pandey42) and Eq.(2.10), the spectral rigidity is

described in terms of µ̄(k;S) as

∆3,µ̄(L) =2

L4

∫ L

0

dx(L3 − 2L2x+ x3)Σ2µ̄(x)

=L

15+

4

L4

∫ L

0

dx(L3 − 2L2x+ x3)

∫ x

0

dS+∞∑

k=0

µ̄(k;S). (2.12)

§3. Properties of limiting level number variance

We analyze the slope of the limiting LNV

g(L) =1

LΣ2µ̄(L) = 1 +

2

L

∫ L

0

dS

+∞∑

k=0

µ̄(k;S), (3.1)

which has the following convergence:

limL→+∞

g(L) = 1 + 2

+∞∑

k=0

µ̄(k; +∞) ≥ 1 + 2µ̄(0; +∞). (3.2)

Since µn(k;S) is monotonically increasing for S ≥ 0, monotonically decreasing for k =

0, 1, 2, · · · , and 0 ≤ µn(k;S) ≤ 1, µ̄(k;S) has the same properties and is bounded by

µ̄(0; +∞) as

0 ≤ µ̄(k;S) ≤ µ̄(0;S) ≤ µ̄(0; +∞) ≤ 1, (3.3)

where µ̄(0; +∞) classifies the cumulative NNLSD (1.2) into three cases: Case 1, the Poisson

distribution if µ̄(0; +∞) = 0; Case 2, the asymptotic Poisson distribution if 0 < µ̄(0; +∞) <

1; and Case 3, the sub-Poisson distribution if µ̄(0; +∞) = 1.

Then, the property of the limiting LNV is evaluated for Cases 1—3 as follows:

Case 1, µ̄(0; +∞) = 0: The limiting LNV Σ2µ̄(L) agrees with the LNV of Poisson statis-

tics: Σ2Poisson

(L) = L. Note that this condition is equivalent to∑+∞

k=0 µ̄(k;S) = 0 since µ̄(k;S)

is monotonically increasing for S and decreasing for k.

Case 2, 0 < µ̄(0; +∞) < 1: Σ2µ̄(L) possibly deviates from Σ2Poisson

(L) in such a way that

the slope of the limiting LNV is 1 at L = 0, increases monotonically with L, and approaches

9

a number 1 + 2µ̄(0; +∞) or more as L → +∞.

Case 3, µ̄(0; +∞) = 1: Σ2µ̄(L) possibly deviates from Σ2Poisson

(L) in such a way that the

slope of the limiting LNV is 1 at L = 0, increases monotonically with L, and approaches a

number 3 or more as L → +∞.

One has Case 1 if the NNLSD of individual components are derived from the scaled

distribution functions ϕn(0;S) as

µn(0;S) = ρn

∫ S

0

ϕn(0; ρnx)dx, (3.4)

where ϕn(0; ρnS) = pn(0;S)/ρn and satisfy∫ +∞

0

ϕn(0; x)dx = 1,

∫ +∞

0

xϕn(0; x)dx = 1, (3.5)

and are uniformly bounded by a positive constant D : |ϕn(0;S)| ≤ D ( 1 ≤ n ≤ N ). Indeed,

the following holds:

|µ(0;S)| ≤N∑

n=1

ρ2n

∫ S

0

|ϕn(0; ρnx)| dx ≤ DSN∑

n=1

ρ2n ≤ DSmaxn

ρn

N∑

n=1

ρn → 0 ≡ µ̄(0;S).

(3.6)

Such a bounded condition is possible when the individual spectral components are sparse

enough.

In general, one may expect Cases 2 or 3 with µ̄(0;S) > 0, which corresponds to strong

accumulation of energy levels, leading to a singular NNLSD of the individual components.

Such accumulation is expected to arise from the symmetry of the system. In the next section,

we will analyze the LNV of the rectangular billiard systems which is known to deviate from

Poisson statistics.

§4. Rectangular billiard system

We present our results on the various statistical measures discussed in the previous section

for a rectangular billiard system whose spectral statistics has been precisely analyzed in a

number of works.1), 5), 11), 15), 18), 19), 31), 32) The eigenenergy levels of this system are given by

ǫn,m = n2 + αm2, (4.1)

where n and m are positive integers, and α is denoted by the lengths of two sides a and b

as α = a2/b2. The unfolding transformation {ǫn,m} → {ǭn,m} is carried out by using the

leading Weyl term of the integrated density of states, N (ǫ), as

ǭn,m = N (ǫn,m) =π

4√

αǫn,m. (4.2)

10

Berry and Tabor observed that the NNLSD of this system possibly deviates from Poisson

statistics when α is rational.1) In this paper, we study irrational cases in addition to a

rational case(α = 1) that are described by a finite continued fraction of the golden mean

(√

5 + 1)/2,

α = 1 +1

1+

1

1+· · ·

1

1+

1

1 + δ= [1; 1, 1, · · · , 1, 1 + δ] , (4.3)

with an irrational truncation parameter δ ∈ [0, 1).

Fig.1 shows the plots of the LNV Σ2(L) for α corresponding to the (a) 25th, (b) 8th,

and (c) 4th approximations of the golden mean, and (d) α = 1. Our analysis is valid in

the region L

µ̃(S) in S → +∞ approaches a number µ̃(+∞) such that 0 < µ̃(+∞) < 1 [plots (b) – (d)

in Fig.4]. Therefore, plots (b)–(d) correspond to Case 2.

For the rectangular billiard in the finite energy region, we have not yet succeeded in

observing Case 3. This case is expected to arise in a square billiard (α = 1) in the high

energy limit where a stronger accumulation of levels is generated. Based on the number-

theoretical result of Landau (1908),43) Connors and Keating have proved that the eigenenergy

levels ǫn,m of square billiard show a logarithmic increase in the mean degeneracy of levels as

ǫ → +∞, which is described as

1−M(+0) ≃4

π

C2√

ln ǫ→ 0, (4.5)

where C2 converges to give C2 ≃ 0.764.11) The above limit corresponds to a delta function

of NNLSD, P (S) = δ(S), and is consistent with µ̄(0;S = +0) = limǫ→+∞M(+0) = 1, which

indicates an extremely slow approach to Case 3 in ǫ → +∞.

Fig.5 shows the cumulative NNLSD M(S) for α = 1. This function is not smooth at the

level spacings separated by a step π/4,1) which correspond to accumulation of levels. Note

thatM(+0) > 0 due to the degeneracy at S = 0. Since µ̃(+0) = M(+0) > 0, this degeneracy

at S = 0 is identified also in Fig.4. As the eigenenergy levels become higher, M(+0) increases

monotonically and approaches 1. In the limit ǫ → +∞ where M(+0) = µ̄(0, S = +0) = 1,

all steps except the step at S = 0 are suppressed since M(S) is monotonically increasing for

S > 0.

Fig.6 shows 1 − M(+0) vs 4C2/π√

ln ǫ for various energy ranges. Although we are

not yet far enough in the high energy region where 1 − M(+0) = 1 − µ̃(+0)

many independent components, and each of which gives an infinitesimal contribution to the

level statistics. Moreover, by developing the approach of Makino et al19) into the statistics

of higher order (long-range) spectral fluctuations, the LNV of systems consisting of infinitely

many spectral components are obtained. The LNV is characterized by the monotonically

increasing functions µ̄(k;S), k = 0, 1, 2, · · · , of the level-spacing S, where the lowest order

term µ̄(0;S) is associated with the NNLSD. The property of the LNV is classified as follows:

Case 1, µ̄(0; +∞) = 0 where the cumulative NNLSD is the Poisson distribution, the LNV is

the Poissonian Σ2Poisson

(L) = L; Case 2, 0 < µ̄(0; +∞) < 1 where the cumulative NNLSD is

the asymptotic Poisson distribution, the LNV deviates from Σ2Poisson

(L) in such a way that

the slope is greater than 1 and approaches a number ≥ 1 + 2µ̄(0; +∞) as L → +∞; Case

3, µ̄(0; +∞) = 1 where the cumulative NNLSD is the sub-Poisson distribution, the LNV

deviates from Σ2Poisson

(L) in such a way that the slope is greater than 1 and approaches a

number ≥ 3 as L → +∞. Therefore, we have shown that deviations from Poisson statistics

(Cases 2 and 3) are possibly observed, not only in the property of NNLSD characterizing

short-range spectral fluctuation as shown in the work of Makino et al ,19) but also in the

property of the LNV characterizing the fluctuations of all ranges.

Note that Cases 2 and 3 may arise when there is a strong accumulation of levels, which is

characterized by the singular NNLSD of individual components. Such accumulation would

be expected when there is a spatial symmetry or time-reversal symmetry. One example is

a rectangular billiard shown in Section 4 where the result shows Case 2 in addition to Case

1, and an extremely slow approach to Case 3 . Similar results due to a spatial symmetry

are also shown in the equilateral-triangular billiard,30), 44) torus billiard,29), 18) and integrable

Morse oscillator.31) Another example is a certain type of systems with the time-reversal

symmetry studied by Shnirelman,33) Chirikov and Shepelyansky,34) and Frahm and Shep-

elyansky.35) In this system, the strong accumulation of energy levels, resulting from the time

reversibility, is reflected in the NNLSD as a sharp Shnirelman peak at small level-spacings.

Rigorous results confirming the Berry–Tabor conjecture (Case 1) are also reported for a

certain classically integrable system. In the work of Marklof13) and Eskin et al ,14) eigenvalue

problems are reformulated as lattice point problems, and it is exactly proved under explicit

diophantine conditions that a two-point spectral correlation exhibits Poisson statistics.

The periodic-orbit theory, applied for an integrable system with degeneracy in orbit ac-

tions, gives a similar result. In the work of Biswas et al ,32) it was shown that the slope gav(L)

of the LNV Σ2(L) = gav(L)L represents the average degeneracy of actions of periodic orbits,

which is greater than 1 for the rectangular billiard with a rational ratio α of squared sides.

Since this property is qualitatively consistent with the property of the limiting LNV (2.10)

13

in Cases 2 and 3, one might expect to have the relation:

gav(L) = 1 +2

L

∫ L

0

+∞∑

k=0

µ̄(k;S)dS, (5.1)

which enables us to discuss the properties of the overall NNLSD from the periodic-orbit

theory. From Eqs.(5.1) and (3.3), we expect that the Case 1 corresponds to the non-

degeneracy of actions: gav(L) = 1, while Cases 2 and 3 correspond to the degeneracy of

actions:gav(L) > 1. Therefore, we have confirmed that Cases 2 and 3 are closely related to

the degeneracy in orbit actions. At the same time, we have only superficially examined the

possibility that the degeneracy of actions, induced by the symmetry, could be an essential

factor for the singularity of individual spectral components that yields Cases 2 and 3. This

part should be investigated in detail in a future work.

There is another approach similar to the one presented in this paper, in which the LNV

is derived in the limit of infinitely many components.45) The LNV is described by the Dyson

two-level cluster function Y2 as

Σ2(L) = L− 2

∫ L

0

(L− S)Y2(S)dS. (5.2)

For spectral superposition, Y2 is described in terms of the cluster function of individual

components y2,n as in Pandey’s work42)

Y2(S) =N∑

n=1

ρ2ny2,n(ρnS). (5.3)

Then, for the overall LNV, one has the convergence

limN→+∞

Σ2(L) = L+ 2

∫ L

0

c̄(S)dS (5.4)

with

c̄(S) = − limN→+∞

N∑

n=0

ρn

∫ ρnS

0

y2,n(x)dx, (5.5)

where c̄(S) = 0 corresponds to Poisson statistics and c̄(S) 6= 0 indicates deviations from

Poisson statistics. Since y2,n is associated with the level-spacing distribution of individual

components as y2,n(x) = 1−∑+∞

k=0 pn(k; x), c̄(S) is rewritten as

c̄(S) = limN→+∞

N∑

n=0

ρn

∫ ρnS

0

+∞∑

k=0

pn(k; x)dx. (5.6)

14

When pn(k;S), k = 0, 1, 2, · · · , are assumed to satisfy the commutation relation∫ ρnx

0

∑+∞k=0 pn(k;S)dS =

∑+∞k=0

∫ ρnx

0pn(k;S)dS, c̄(S) is described as c̄(S) =

∑+∞k=0 µ̄(k;S) and limit (5

.4) is consistent

with the limit (2.10).

Acknowledgment

H.M. is grateful to Prof. M. Robnik, Prof. A. Shudo, and Prof. Y. Aizawa for helpful sugges-

tions. This work is partially supported by KAKENHI(No.18740241 to H.M., No.17540100 to

N.M., No.17540365 to S.T.), and by a Grant-in-Aid for the “Academic Frontier” Project at

Waseda University from the Ministry of Education, Culture, Sports, Science and Technology

of Japan.

Appendix A

Derivation of Eqs.(2.3) and (2.4)

We briefly show the detailed process to derive Eqs.(2.3) and (2.4) from Eqs.(1.4), (1.7),

and (1.8). First, we rewrite Eq.(1.3) as

+∞∑

k=0

(k − L)2EN (k;L) =+∞∑

k=0

k2EN(k;L)− L2, (A.1)

where we have used the normalization conditions∑+∞

k=0EN(k;L) = 1 and∑+∞

k=0 kEN (k;L) =

L, which follow from Eqs.(1.7) and (1.8), and the relations k =∑N

n=1 kn and∑N

n=1 ρn = 1

as+∞∑

k=0

EN(k;L) =N∏

n=1

+∞∑

kn=0

en(kn; ρnL) = 1, (A.2)

+∞∑

k=0

kEN(k;L) =

N∑

n=1

+∞∑

kn=0

knen(kn; ρnL)

N∏

i=1,i 6=n

+∞∑

ki=0

ei(ki; ρiL)

=N∑

n=1

ρnL · 1N−1 = L. (A.3)

By using Eqs.(A.2) and (A.3),∑+∞

k=0 k2EN(k;L) in the right hand side of Eq.(A.1) is calcu-

lated in a similar manner:

+∞∑

k=0

k2EN (k;L)

=

N∑

n=1

+∞∑

PNi=1,i6=n ki=0

(

N∏

i=1,i 6=n

ei(ki; ρiL)

)

+∞∑

kn=0

k2nen(kn; ρnL)

15

+

N∑

n 6=i

+∞∑

kn=0

+∞∑

ki=0

knen(kn; ρnL)kiei(ki; ρiL)

=N∑

n=1

(

+∞∑

kn=0

k2nen(kn; ρnL)

)(

+∞∑

ki=0

ei 6=n(ki; ρiL)

)N−1

(A.4)

+

(

N∑

n=1

+∞∑

kn=0

knen(kn; ρnL)

)2

−N∑

n=1

(

+∞∑

kn=0

knen(kn; ρnL)

)2

=N∑

n=1

+∞∑

kn=0

k2nen(kn; ρnL) + L2 −

N∑

n=1

(ρnL)2. (A.5)

Since the off-diagonal terms n 6= i vanish in the second equality, we have

+∞∑

k=0

k2EN(k;L)− L2 =

N∑

n=1

(

+∞∑

kn=0

k2nen(kn; ρnL)− (ρnL)2

)

=

N∑

n=1

+∞∑

kn=0

(kn − ρnL)2en(kn; ρnL). (A.6)

Therefore, the overall LNV of the system with N independent components is described as

Σ2N(L) =N∑

n=1

σ2n(ρnL), (A.7)

where σ2n is the LNV of the spectral component:

σ2n(ρnL) =+∞∑

k=0

(k − ρnL)2en(k; ρnL). (A.8)

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18

FIG.1 Level number variance Σ2(L) of the rectangular billiard systems for (a) 25th, (b) 8th,

(c) 4th approximations of α = (√

5 + 1)/2, and (d) α = 1. The solid line denotes the LNV

of Poisson statistics, Σ2Poisson

(L) = L. We used energy levels ǭn,m ∈ [4000× 107, 4001× 107].

The total numbers of levels are (a) 10000457, (b) 10000428, (c)10000266, and (d) 10000162.

The truncation parameters δ are (a) π × 10−9 (b) π/3× 10−7 and (c) π × 10−9.

FIG.2 Slope of the level number variance, g(L) = Σ2(L)/L, for (a) 25th, (b) 8th, (c) 4th

approximations of α = (√

5 + 1)/2, and (d) α = 1. The solid line, g(L) = 1, corresponds to

Poisson statistics.

19

FIG.3 Function − ln [1−M(S)] for (a) 25th, (b) 8th, (c) 4th approximations of α = (√

5 +

1)/2, and (d) α = 1. The dotted line corresponds to the cumulative Poisson distribution

M(S) = 1− e−S.

20

FIG.4 Parameter function µ̃(S) for (a) 25th, (b) 8th, (c) 4th approximations of α = (√

5 +

1)/2, and (d) α = 1. The solid line, µ̃(S) = 0, corresponds to the Poisson statistics.

FIG.5 Cumulative NNLSD M(S) for α = 1. We used 999844 levels ǭn,m ∈ [40×106, 41×106]

for plot (A), 10001872 levels ǭn,m ∈ [4000000× 107, 4000001× 107] for plot (B). We observe

M(+0) ≃ 0.767 for plot (A) and M(+0) ≃ 0.826 for plot (B). The dotted curve is the

cumulative Poisson distribution M(S) = 1− e−S

21

FIG.6 Numerical test of Eq.(4.5) for a square billiard (α = 1). The solid line exhibits the

theoretical prediction, which is valid in the semiclassical limit ǫ → +∞. In each plot, we

used 107 eigenenergy levels obtained from the unfolded energy range. The error bar exhibits

the energy range acquiring numerical data.

22

Introductionlimiting level number varianceProperties of limiting level number varianceRectangular billiard systemConclusion and Discussion Derivation of Eqs.(2.5ex.3) and (2.5ex.4)